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11 12 1
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E1 E2 E3
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LC-() (. 2.52).
3
10 10 L
=
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( )
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2.14 2.13 -
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1001 1001
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. 2.57
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. 2.59
MathCAD :
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h tm h h tm hh
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. 2.62
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2.1
R R1 1000R R= 2 100 = 3 10R R=
-
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2.2
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15.
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10 0.005 30.7 10 50
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2. -MathCAD
Simplify. . 2.4
2.4
2.1.
1 2 2a b
a b
+
2 2 2a b
a b
3 3 2 23 3a b 3b a b
a b
a+ +
+
+
4 3 23a b a 2 33 b a b+
a b
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6 4 3 2 2 3 4
3 2 2 3
4 6 4
3 3
a b a b a b a b
a b a b a b
+ + + +
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cos sinx x+ 8 2 3 4
2 5x x
4x
x
+
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10 ( ) ( ) ( )a b a b a b+ + +
.2. -Expand. . 2.5.
2
2.5.
1 ( ) ( ) ( ) ( )1 1 2x x x x 2+ +
2 ( ) ( ) ( ) ( )3 3 2x x x x 2+ +
3 ( ) ( ) ( ) ( )1 1 4x x x x 4+ +
4 ( ) ( ) ( ) ( )1 1 3 3x x x x+ +
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( ) ( ) ( ) ( )4 4 2 2x x x x+ + 5
2.5.
6 ( ) ( ) ( ) ( )1 1 5x x x x 5+ +
7 ( ) ( ) ( ) ( )5 5 2 2x x x x+ +
8 ( ) ( ) ( ) ( )5 5 3x x x x 3+ +
9 ( ) ( ) ( ) ( )5 5 4 4x x x x+ +
1 ( ) ( ) ( ) ( )1 2 3x x x x 4+ + 0
2 - . . 2.6.
2.6
.3. Factor
1 4 213 36x x +
2 04 3 23 15 19 3x x x x+ +
3 4 3 27 6x x x x + +
4 4 210 9x x +
5 x4 217 16x +
6 x4 2
x 26 25 + 7 x x +4 229 100
8 4 240 144x x +
9 4 245 324x x + 10 4 3 211 9 18x x x x + +
2 , -Collection Subexpression. . 2.7.
2.7.
.4.
1 2 2 2 22x a y x y x x x y + +
2 ( )2 2 2 23 2a y x x y x x x y + + +
3 ( )2 2 2 23 2x a y x x y x + +
4 ( )2 2 2 22 2x a y x y x x x y x x + + +
5 ( ) ( )2 2 2 23 2 2x a x x y x x y x + + +
108
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6 2 2 2x a y x x x y +
2.7.
7 ( )2 2 2 22
a ya y x y x x x yx
+ +
8 ( )2 2 2 22a y x y x x a y x y + + +
9 ( ) ( )2 2 2 22 1x a y x y x x x y + + + + 1
1 ( ) ( )2 2 2 22 2 3a y x y x x x y + + + 0
2.5. , Coeffs. . 2.8.
2.8
1 ( ) ( ) ( ) ( )x a x b x c x d+ + + +
2 ( ) ( ) ( ) ( )x a x b x c x d+ + +
3 ( ) ( ) ( ) ( )x a x b x c x d+ + +
4 ( ) ( ) ( ) ( )x a x b x c x d+ + +
5 ( ) ( ) ( ) ( )x a x b x c x d + + +
6 ( ) ( ) ( ) ( )x a x b x c x d + +
7 ( ) ( ) ( ) ( )x a x b x c x d+ +
8 ( ) ( ) ( ) ( )x a x b x c x d+ +
9 ( ) ( ) ( ) ( )x a x b x c x d +
1 ( ) ( ) ( ) ( )x a x b x c x d 0
2 . . 2.9.
2.9.
.6.
1 ( )2 2
dx y
dx+
2( )
22
2
da x b x c
dx + +
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3( ) ( )( )
22
2ln sin 2 1
dx x x x
dx + +
2.9.
4( )
( )
2 2
2ln 2
sin
xd xx x edx x
+ +
5
( )
3 32
32
d xx x
dx tg x
+ +
6( )
22
2
da x b x c
dx + +
7
( )
42
4
1
ln
dx b x cdx x
+ +
8 ( )2 2x xd
e edx
+
9( )
22 2
2
x xd e edx
+
10 ( )( )3
sin 2
3
x xd e edx
+
2 . . 2.10.
2.10.
.7.
1 ( )2 2 2y z dx dy dz+ + 2
( )2 1b
a
dx+
3 4b
adx
4 ( )
( )
3
5
sin
cos
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x
5 ( )
( )4
sin
cos
xdx
x
6 2
41
ydy
y
110
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7 x x
x x
e edx
e e
+
8 xe x dx
2.10.
9 ( )ln x dx 10 2
0
hr x
dxh
2 x . . 2.11.
2.11
.8.
1
cosx
t dt a( )0
=
2 2a x b x c + =
3 21
385x
i
i=
=
4 3 2 0a x b x c x + + =5 ( ) ( )sin cos 0x x+ =
6 0k x b y + =
7 2 0k x b y + =
8 43
10xe x+ + =
9 ( )1
0tg xx
+ =
10
( )
1 1
ln 2x x
+ =
2.9. f1(x) y(x),Substitute. y(x)f1(x). 2.12.
111
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2.12
1
( )( ) ( )( )
( ) ( )
22 2
cos sin
cos 2! sin 3!
x xy x
x x
+=
+ , =( ) ( ) ( )
2 2
1 cos sin 1f x x x +
2( )
( )( ) ( )( )
2 21 cos
sin
tg x xy x
x
+ = , ( ) ( )
( )2
1 2
11
cosf x tg x + =
3 ( ) ( )( ) ( )( )
2 2
1 sin
cos
ctg x xy x
x
+ = , ( ) ( )( )
2
1 2
11
sinf x ctg x + =
4( )
( )
( ) ( )
2
3
2cos
sin
tg xy x x
x= , ( ) ( )
( )
( )1
sin
cos
xf x tg x
x =
5( )
( )
( ) ( )
2
2
3sin
cos
ctg xy x x
x= , ( ) ( )
( )
( )1cos
sinf x ctg x =
6( )
( )
( ) ( ) ( ) ( )
sin
sin cos sin cos
x yy x
y y
+=
+ x,
( ) ( ) ( ) ( ) ( ) ( )1 sin co sin cos sinsf x x y y x x y + = +
7( )
( ) ( ) ( ) (
( ) ( )
)2 2 2
2 2
4 cos 4 sin cos sin
cos sin
2x x x
y xx x
+ =
+
( ) ( ) ( )2 2
1 cos sin 1f x x x + =
x,
8( )
( ) ( ) ( ) ( )( )
2 2 2cos 2 sin sin
cos
2x x x tg
y xx
+ + =
( ) ( ) ( ) ( ) ( )
x,
( )
2 2 2 2
1 2
1cos 2 sin sin
cosf x x x x tg x + + =
9
( )
( )
( )
( )
( )
( )
2 2
2 2cos
sin 3 cos 3
sin
8 cos 2
x
xy x
x
=
x,
( ) ( )
( )
( )
( ) ( )
2 2
1 2 2
sin 3 cos 38 cos 2
sin cos
x xf x x
x x
=
112
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( )( ) ( )( )
( )
22 2
cos sin
cos
x xy x
x
+= , =( ) ( ) ( )
2 2
1 cos sin 1f x x x +
10
2 f(x)x=0 9 . - x=x0. f(x) . 2.13.
2.13
.10. x0
1 f(x)=ex, x0=52 f(x)=e
2x, x0=2
3 f(x)=cos(x), x0=/44 f(x)=sin(x), x0=/25 f(x)=sin
2(x), x0=/2
6 f(x)=cos2(x), x0=/4
7 f(x)=tan2(x), x0=/48 f(x)=sin3(x), x0=/49 f(x)=arcsin(x), x
0=/4
10 f(x)=arcsin2(x), x0=/4
2.11. x , . 2.14, -Parfrac.
2.14.
14
1
1
x
2 2
3 2
2 3 1
2 9 1
x x
x x x
+
8+
34
1
1
43 2
1
2 9 1
x
8x x x
+
+
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5 2
3 2
1
2 9 1
x
x x x
8+
6 4
3 2
1
2 9 1
x
x x x
8+
7 2
4
1
1
x
x
2.14.
8 3
4
1
1x
9 2 3
4
1 x
1
x
10 2 3
1 2 1x
x
2.12. , . 2.15.
2.15
1 24 1
lim2 1
x
x12
x
2 24 1li
x
6m
2 1x x
3 ( )( )0
lim 1 cosx
x
4 1lim
!n n
5 1lim 1
n
n n
+
6 1limx
7 ( )0
sin 2limx x
8 ( )0x
sinlim
9 ( )
( )0sin 2
limsin 5x x
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10 ( )0
1 coslimx x
3.
3.1. , . 2.16.
2.16
1
1
a c e
f
1 1 1
1 1b d
2 ( )a b
3 ( )a b c d
4
a b
c d
5
6 8
b d
f h
1
e
2 3
g
4
5 7
a c
6
a
b
c
7
5 6
1 2
3 4
8 1
4 5 6
2 3
9
1 3
2 4
a
b
10
1
2
3
3. 72. , . 2.1 .
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2.17.
1
c d
1 2
3 4
a b
2.17.
2 a a
a a
1 2
8 4
3 1 216 4
a aa a
4 21
32 4
a
a a
a
5 2
a a
1
64 4
a a
6 2
4a a
1
4
a a
7 4
a a
a
1 2
2a
8 1 2a a
8 16a a
9 1 2
8 1
a a
a a
10 10 100
1 10
a a
a a
3.3.
6 10
a
b
c
.
Substitute, c 18
10 5 1
7 10 4
2
a, b . 2. .
2 .18.
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1 b 1a=1, =1, c=
2 , b=1, c=a=1/2 1
3 /a=1, b= , c=11 2
4 , b c=1a=2 =1,
5 ,a=1 b 2, c== 1
6 a 1, b c=2= =1,
7 a=2, b=2, c=1
2.
8 b=2 =2
18.
a=1, , c
9 2a=2, b=1, c=10 a=1, b=4, c=1
.
.1. -
, . -. E .
2.19.
2.19
4
4 -
r
1 E=110 , r=1 2 E=220 , r=2 3 E=110 , r=2 4 E=220 , r=1
5 E=440 , r=1 6 E=440 , r=2 7 E=110 , r=3 8 E=110 , r=4 9 E=220 , r=3 10 E=220 , r=4
5.
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5.1. -, , (. 2.100). . 2.20.
. 2. 100.
2.20
-
1, R
2,
3, R
1,E 2,E 3,
1 20 4 5 20 2 1
2 20 4 5 10 4 2
3 50 10 5 10 4 24 50 10 5 20 4 2
5 10 20 10 20 4 2
6 40 40 20 20 4 2
7 80 20 10 20 4 2
8 5 5 10 20 4 2
9 5 5 10 40 4 2
10 5 5 10 40 10 5
5.2. -, , (. 2.101). -. 2.21.
:j=(-1)1/2
.
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1Z
2Z
3Z1I 3I
3E1
2I2
. 2.101.
-
2.21
1,Z
2, Z
3,Z
1,E 2,E 3,E
1 20+j20 4-j4 5+j5 j20 2 j12 20+j20 4-j4 5+j5 j10 4 j23 50+j50 10-j10 5+j5 j20 4 j24 10+j10 4-j4 5+j5 j20 4 j2
5 10+j10 20-j20 10+j10 j20 4 j26 40+j40 40-j40 20+j20 j20 4 j27 80+j80 20-j20 10+j10 j20 4 j28 5+j5 5-j5 10+j10 j20 4 j29 5+j5 5-j5 10+j10 j40 4 j2
10 5+j5 5-j5 10+j10 j40 10 j5
6.
6.1. , -, - -:
.. 2.22
. 2.39.
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2.22
-
,
f
p 1,
,K
x 2
,pR
,U
1 2 5 0.7 220 50 22 2.5 5 0.5 220 50 2
3 2 5.5 0.5 220 50 2
4 3 5 0.5 220 250
5 3 5.5 0.7 220 250
6 2 6 0.7 220 50 2
7 2 6 0.5 220 50 2
8 2.5 6 0.7 220 50 2
9 3 6 0.7 220 50 210 3.5 6 220 50 20.7
7.
7.1. -LC-() -
,
(. 2.102).
1 10R R= 2 100R R= , 3 1000R R= . -. 2.23.7.2. -
, ,
-LC- . -
1 10R R= 2 100R R= 3 1000R R= .
. () , , (), - (), - (), - ().
-.
. . 2.23.7.3.
-LC-
-
-: -
1 10R R= , 2 100R R ,= 3 1000R R= .
. -
-
. -. 2.23.
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. 2.102.2.23
,L ,C , R 1 900 800 12 850 850 1
3 800 900 1
4 750 950 1
5 700 1000 1
6 900 1000 1
7 850 950 1
8 800 900 1
9 750 850 110 700 800 1
. -. -. 2.24.
8..
8.1. - . 24 48 .
.
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2.24.
1 -
--
--
2 220U = ( )U t
T
U
t
50f =
1 2U U R
TVD
( )U t
2 -
-- -
2 220U =
( )U t
T
U
t
50f =-
1U 2U R
T VD
( )U t
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2.24.
3 - ----
-
=90
2 220U = ( )U t
T
U
t
150f
T= =
=90
1U 2U R
TVS
( )U t
4 -
-
-
-
-
=90
2 220U = ( )U tU
t
150f = =
TT
=90
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RTVS
( )U t
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2.24.
5
-
-
-- -
-
- -
- -
=90
2 220U = ( )U tU
t
150f = =
TT =90
T
1U U2 R
VS
( )tUVD
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2.24.
6 - --- --
-
-
=90
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U
t
150f
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=90
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T1
VS
( )U t2VS
7
-----
--
-
-
=0.5
2 220U = ( )U tU
T
t
150f
T= =
0.5 =
1800f
T= =
T
1U 2UR
T VD( )U t
1VT
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2.24.
8 --
--- --
-=90
2 220U = ( )U tU
t
150f
T= =
T
1U 2U
RT
1VS
( )U t
2VS 2VD
1VD
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-
- --
-
2.24.
9 -
=90
2 220U = ( )U tU
T 150f
T= =
1BU 2BU
2T2VD
( )U t1
3T
CU 2CU
1AU 2AU
1T1VD
3VD
2.24.
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10 -
-
--
2 220U = ( )U t
T
U
150f = =
T
-
1BU 2BU
2T2VS
( )U t1U C 2CU
3T
1AU 2AU
1T1VS
3VS
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9.
- 100 . ,, --(). : -. : L
-
9.1. --LC-
1 2L L= = , 1 2C C C= = , 1 2R R= = 1 10
. - R R= ,
. . 2.23.2 100R R= ,
3 1000R R=
. 2.103. -LC-
10.
10.1. . - ,-. -. 2.25.
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2.25.
1 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
210 355 455 555 755 1005 1385 2005 3005 4005 5205
2 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
215 365 465 565 765 1065 1395 2015 3015 4015 5215
3 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
220 370 470 570 770 1070 1400 2020 3020 4020 5220
4 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H, 240 390 490 590 790 1040 1420 204/
0 3040 4040 5240
5
B, 0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H, 250 400 500 800 1050 1430 2050 3050 4050 5250600/
6 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
210 360 460 560 760 1010 1390 2010 3010 4010 5210
7 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
225 375 475 575 775 1025 1405 2025 3025 4025 5225
8 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
235 385 485 585 785 1035 1415 2035 3035 4035 5235
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2.25.
9 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
245 395 495 595 795 1045 1425 2045 3045 4045 5245
10 B,
0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.55 1.6
H,/
255 405 505 605 805 1055 1435 2055 3055 4055 5255
11.
11.1. . kb knz. - . -kb=f(knz) . 2.26.
2.26.
-
1 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.119 1.101 1.081 0616 1.054 1.0452 1.041 1.0381.0706 1.2 k 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3nz
kb 1.12 1.102 1.087 1.0709 1.0617 1.055 1.0453 1.042 1.0393 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.121 1.103 1.088 1.071 1.06 1.056 1.0453 1.04318 1.04
4 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.117 1.099 1.084 1.0706 1.0614 1.052 1.045 1.039 1.0365 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.118 1.098 1.083 1.0704 1.0611 1.049 1.0448 1.038 1.035
2.26.
6 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb
1.118 1.1 1.085 1.0707 1.0615 1.053 1.0451 1.04 1.037
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7 3knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75
k 1.117 1.097 1.082 1.0704 1.061 1.05 1.045 1.038b 1.0368 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb
1.108 1.09 1.075 1.0697 1.0605 1.043 1.0441 1.03 1.0279 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.109 1.091 1.076 1.0698 1.0606 1.044 1.0442 1.031 1.02810 knz 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
kb 1.110 1.092 1.077 1.0699 1.0607 1.045 1.0443 1.032 1.029
12.
MATHCAD
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) ,
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) *.txt.
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3
MICROSOFT OFFICE EXCEL
1. EXCEL
Excel Microsofice
Excel . - -
.
, : / / (. 3.1). Excel.
ft Of--
.
Excel
. 3.1
3 -Excel.
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2.1.
2.
, -(. 3.2). :
0 0.05L = ,
,
6160 10 R = 632 10 r =
160= -.
i
2 r
2R
. 3.2
w, 0L .
2.2.
-
L i
Li
= . (3.1)
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w S
w = . (3.2)
2S r= . (3.3)
i w H l = , (3.4)
H l
iw
= , (3.5)
H , l ( )2 l R r= . (3.6)
-
B S
=B S . (3.7) B
H
0 B H= , (3.8)
70 4 10 = .
(1) (7)
( )( )
2 222 0 0
2
H w rw w S B S L w w
H li H l l H R
w
r
= = = = =
( )( )
2 20 w rL w
2 R r
= . (3.9)
( ) 0 0L w L = . (3.10)
.
( ) ( ) 0w L . (3.11)L w= ( )f w ,
w , .
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, -
( )
( )
( )
2 2 2 2 120 32 10 160 4 10w r wL
7
0
6
0.05
2 2 10 160 32
f w
R r
= =
( ) 2 108.042 10 0.05f w w = .(3.12)
2.3. Exce
l
w w 7880= = (3.12) B3. C3
-
7881, w=1.
. (3.11) (. 3.3). C3
-6.36835E-05. B4
B3-B13(. 3.4 ).
. 3.3
B3-B4. -. -
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C3. . - C3-C13 (. 3.4 ). .
B3-C13. Excel (. 3.5 ). - (.3.5 ) , -(. 3.6).
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, .
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. 3.6
2.4. .
E
r (. 3.7). VD1. I.
r
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r= 1 ,
-
: E = 1 , T= 300 , I0= 1010
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2.5.
E-r
VD1
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r
= .
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0 1
U q
k TI I e
=
, (3.14)
0 () ,p-n-U , p-n-,
191.6 10 q = ,
23 1.38 10
k
= ,
T p-n-.(12) (13)
0 1
.
U q
k TI I e
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18 17 38.6471.3810 300
10 10 1 10 1
11 .
1
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2.6. Excel
(3.17) - -
.
U
( ) ( )( )
17 38.6471 10 1
Uf U e
2 1 .f U U
=
= (3.17)
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1-2 -
15 (. 3.8).
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B1 ( )1U , C1
( )2f U (. 3.9).
141
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. 3.9
B1 B2, C1 C2 ( - , - ). -A1-C2 C15.
, B1-B15
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. 3.10
A1-C15 , (. -, 0.9-0.96,.
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( )1f U
( )2f U (. 3.11).
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. 3.11
0.94 U= 0.06 I=
-. , .
3.
3.1.
(. 3.12), -.
0.94 0.06 0.0564 P= =
:
1 10 E = , 2 5 E = , 3 3 E = :
, ,1 1R = 2 2 R = 3 4 R = ., .
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E1 E2 E3
R1
R2
R3I3
. 3.12.
I1 I2
I1, I2, I3
2
1 1 1 2 2
1 1 1 3 3 3
I R I R E
I R I R E
1 2 3
I I I
= + +
= + + (3.18)
0
0
I R I R I E E
I R I I R E E
+ + = + + = .
(3.19)
-
4
= +
1 1 2 2 3 1 2
1 1 2 3 3 1 3
1 2 31 1 1 0I I I =
(3.19)
1 2
1 3
0 1 2 0
0 1 0
1 1 1 1 1 1
R R
A R R
= =
10 3 7
0 0 0
B E E
(3.20)
1 210 5 5E E
1 3
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.
(3.21)
3
X I A B = =
(3.18) 1
1
I
2
I
(3.20) A1-C3, (3.21) E1-E3 (. 3.13).
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. 3.13
-G1-I3. (. 3.14).
. 3.14
A1:C3 (. 3.15).
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. 3.15
Excel , -. F2, -++. , 0,285714. (. 3.16).
. 3.16
G1-I3, , E1-E3.
F14-F16,
,
(.3.17).
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. 3.17
G1-I3 E1-E3 (. 3.18). F2, -++.
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147
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(. 3.19)
3
2,428571
1,142857
I
I
(3.23)
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1
21,285714I = ,
. 3.19
(3.18).
3.2.
Excel. A1-C3, E1-E3 (. 3.13). A. A5. (. 3.20).
. 3.20
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-A1-C3 (. 3.21).
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. 3.23
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, G1-I3, A7. - , J1-L3, B7.
, M1-O3, C7. I1 , A7 A5. A8. I2 , B7 A5. B8. I3 , -C7 A5. C8 (. 3.24).
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.
3.3. , -,
-(. 3.25).
L 1R
C 2R
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. 3.25
150
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Excel , -.
( )( )3
Cdu t
C idt
= t . (3.24)
(3.24)
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( ) ( )
( )3
1CC
U pZ p
I p C p= =
. (3.26)
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( )3
1 1CC
U jZ j j
I j C j C
= = =
,
(3.27)
2 f = ,
f .
50f = , -.
( )CY j C j = . (3.28)C 2R
( )22
1CR
Y j C jR
= + . (3.29)
C 2R
( )21
1CRZ j
C j2
= +
. (3.30)
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( ) 1
2
1
1Z j R j L
C j
R
= + +
+.
(3.31)
(3.31), -
( ) 1
2
2 2
21
2 2
2
2
21
2 2 2 22 2
2 2
.
1 1
R CR j L
C CR R
= + +
+ +
1
1 1
1
1
1
Z j R j L
C jR
C j C jR R
C jR
R j L
CR
= + +
+ =
+
= + + =+
(3.32)
, -(3.32)
( )
2 2
2
2
21
1
arctg 1
1
CR
RR
2 2
L
C2
2R
C
+
=
+
.
+
(3.33)
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(3.32)
2
1
2
2 2
2 2
2 2
1C
R R
+ ( ) 21
2 21M
R CZ R L
C
= + + +
-
.
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